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Permutations of the positive integers avoiding arithmetic progressions oflength $5$ were constructed in (Davis et al, 1977), implying the existence ofpermutations of the integers avoiding arithmetic progressions of length $7$. Weconstruct a permutation of the integers avoiding arithmetic progressions oflength $6$. We also prove a lower bound of $\frac{1}{2}$ on the lower densityof subsets of positive integers that can be permuted to avoid arithmeticprogressions of length $4$, sharpening the lower bound of $\frac{1}{3}$ from(LeSaulnier and Vijay, 2011). In addition, we generalize several results aboutforbidden arithmetic progressions to construct permutations avoidinggeneralized arithmetic progressions.

Forbidden arithmetic progressions in permutations of subsets of the integers